then it is not a session cookie. you say you understand, but i don't think you do. a session cookie will not be returned to the server in a different session. it's deleted.
a cookie that is not a session cookie (eg, a cookie which expires in 2021) is a normal cookie which the webview will return to the server like any other normal cookie.
do you know for a fact exactly what the server is returning? have you used okhttputils2 to look at the cookies?
also, it sounds like you are connecting to the website with your desktop browser. you then seem to say that you want to take the cookie from your desktop and use it in a webview? that's pretty wild. in any case, if a session cookie is involved, it's not going to work. (even if it's a normal unexpired cookie, it might not work; it depends on exactly what's in the cookie.)
so, if the cookie is not a session cookie, is there some reason you simply can't log in to the site with your webview and try to connect again with the webview to see if it works? i have no problem sending cookies back to the server with a webview. it happens automatically.